All Questions
12
questions
1
vote
3
answers
60
views
Proving that $f(x) = 6\ln(x^{11}-4) -2$ is one-to-one
Please verify my proof, and if there are any mistakes please explain.
Prove that that this function is one-to-one: $f(x) = 6\ln(x^{11}-4) -2$.
Suppose $f(x_1) = f(x_2)$
$\implies 6\ln(x_1^{11}-4) -2 =...
2
votes
2
answers
121
views
$a = \log_{40}100, b = \log_{10}20$.How can I express $b$ depending only on $a$?
Let $a = \log_{40}{100}, b = \log_{10}{20}$. How can I express $b$ depending only on $a$? I tried using the formula to change the base from $40$ to $10$, but couldn't get it just depending on $a$.
I ...
- 95
2
votes
1
answer
64
views
Finding solution set of $\frac{1}{\log_4 \left(\frac{x+1}{x+2}\right)}<\frac{1}{\log_4(x+3)}$ without using derivatives
$$\frac{1}{\log_4\left(\frac{x+1}{x+2}\right)}\lt \frac{1}{\log_4(x+3)}$$
This inequality can be solved by using the monotonicity of $f(x)$ on $x\in(-1 ,\infty)$ where $f(x)=\frac{1}{\log_4\left(\...
- 6,411
3
votes
2
answers
48
views
Is there a gap in my proof? If $a>1$ and $\log_ab_1>\log_ab_2$, then $b_1>b_2.$
I need to prove that
If $a>1$ and $\log_ab_1>\log_ab_2$, then $b_1>b_2.$
My attempts:
Let $\log_ab_1=x, \log_ab_2=y$ we have $\begin{cases} a^x=b_1 \\ a^y=b_2 \end{cases} \Longrightarrow a^...
user548054
0
votes
1
answer
64
views
Eliminating logs in Big-Oh proofs
I am trying too prove:
log2(9n) is BigO(log2n)
I know this is achieved by Big-O forumla/proof f(n) <= C*g(n) where C is a constant. So far, I have:
...
- 103
2
votes
5
answers
458
views
Write $2+\log_3 x+\log_9 x^4-\log_{27} x^5$ into one single logarithm
I'm trying to answer this problem:
Write $$2+\log_3 x+\log_9 x^4-\log_{27} x^5$$
into one single logarithm.
This is what I'm stuck with:
$$\frac{\log x}{\log 3}+ \frac{\log x^4}{\log 3^2}+\frac{\...
- 37
3
votes
2
answers
134
views
Why can't I find the value of $x$ using logarithms?
This is concerning a question in stack exchange : Sum of real values of $x$ satisfying the equation $(x^2-5x+5)^{x^2+4x-60}=1$. I was actually wondering why the correct result is not obtained when ...
- 1,339
3
votes
3
answers
347
views
Simplify the expression: $a^{\log {\sqrt \frac bc}}×b^{\log {\sqrt \frac ca}}×c^{\log {\sqrt \frac ab}}$
My problem is
Simplify the expression:$$a^{\log {\sqrt \frac bc}}×b^{\log {\sqrt \frac ca}}×c^{\log {\sqrt \frac ab}}$$
Here $a,b,c \in \mathbb {R^+}$
My way:
$$\begin{cases}
\frac bc=e^x ...
- 429
4
votes
2
answers
138
views
Is my method correct for to prove $a^{\log_b c} = c^{\log_b a}$?
My problem is to prove this equality:
$$a^{\log_b c} = c^{\log_b a}$$
My method:
$$\begin{cases}
\log_b a=m\\
\log_b c=n\\
\end{cases} \Rightarrow \begin{cases}
a=b^m\\
c=b^n\\
\end{cases}
\...
- 429
1
vote
1
answer
571
views
Simple proof that $\lfloor \log_2 k \rfloor + 1 = \lceil \log_2 (k + 1) \rceil$
Is there any way to show for $k \in \mathbb{N}$
$$\lfloor \log_2 k \rfloor + 1 = \lceil \log_2 (k + 1) \rceil$$
without casework, or of little of it as possible? I've tested it for some integers and ...
- 353
1
vote
3
answers
6k
views
Proof that $\log_a b \cdot \log_b a = 1$
Prove that $\log_a b \cdot \log_b a = 1$
I could be totally off here but feel that I have at least a clue.
My proof is:
Suppose that $a = b$, then $a^{1} = b$ and $b^{1} = a$ and we are done.
...
- 153
2
votes
2
answers
170
views
Proof of a logarithmic equation
If \begin{align}\log_{16}{15} &= a\\
\log_{12}{18} &= b\\
\log_{25}{24} &= c\end{align}
then prove that $$c=\frac{5-b}{2(8a - 4ab -2b +1)}$$
My attempt: I tried to prove it by applying ...
- 1,319